I have a theory concerning Rapture in Bioshock infinite: SPOILERS AHEAD!
I think it is possible that Robert Lutece comes from a universe where Rapture exists, and that world should be the oppostite of the Columbia universe - so Robert is a man instead of a woman, the city in the sky is the city in the sea. This is NOT Booker's universe. Booker's version in this universe is female and possibly Russian (opposite of American), and Anna is her son named Andrei (Andrew Ryan).
So there is my theory to why is Booker able to control the Battysphere.
I think it is possible that Robert Lutece comes from a universe where Rapture exists, and that world should be the oppostite of the Columbia universe - so Robert is a man instead of a woman, the city in the sky is the city in the sea. This is NOT Booker's universe. Booker's version in this universe is female and possibly Russian (opposite of American), and Anna is her son named Andrei (Andrew Ryan).
So there is my theory to why is Booker able to control the Battysphere.